Kondo quasiparticle dynamics observed by resonant inelastic x-ray scattering

Effective models focused on pertinent low-energy degrees of freedom have substantially contributed to our qualitative understanding of quantum materials. An iconic example, the Kondo model, was key to demonstrating that the rich phase diagrams of correlated metals originate from the interplay of localized and itinerant electrons. Modern electronic structure calculations suggest that to achieve quantitative material-specific models, accurate consideration of the crystal field and spin-orbit interactions is imperative. This poses the question of how local high-energy degrees of freedom become incorporated into a collective electronic state. Here, we use resonant inelastic x-ray scattering (RIXS) on CePd3 to clarify the fate of all relevant energy scales. We find that even spin-orbit excited states acquire pronounced momentum-dependence at low temperature—the telltale sign of hybridization with the underlying metallic state. Our results demonstrate how localized electronic degrees of freedom endow correlated metals with new properties, which is critical for a microscopic understanding of superconducting, electronic nematic, and topological states.

The input values for the background bandwidth W , the hybridization constant V , and the f -level energy E f are not obtained from first-principles calculations, nor do they necessarily reflect values taken from experiment. Rather, the calculation remains valid for the given experimental measurements as long as the universality of the Kondo physics holds, where the given quantities are universal functions of T /T K or E/(k B T K ). Since T K depends exponentially on the ratio (W E f /V 2 ), the given parameters will reproduce the data as long as this ratio is approximately correct. To illustrate these characteristics, in Supplementary Figure 1, we show a comparison of calculations for two sets of parameters. Goremychkin et al. have calculated the dynamic magnetic susceptibility of CePd 3 as the Lindhard susceptibility of a DFT+DMFT simulation, taking into account two-particle vertex corrections Γ loc irr [5]. In Supplementary Figure 2, we present a comparison of these results with RIXS spectra obtained at similar momentum transfers and temperatures (cf. colored lines/data adopted from the Supplementary Materials of Ref. [5]). For reference, we also show the results of an earlier DFT+DMFT calculation by Sakai [6] (dashed line), as well as our AIM/NCA calculation, as discussed above (dot-dashed line).
The direct comparison between RIXS spectra and three computational models of χ reveals some interesting similarities and differences. As in our simpler AIM/NCA model, χ obtained by DFT+DMFT is dominated by excitations within the J = 5/2 ground state manifold and features only a weak broad signal at the spin-orbit energy. However, the DFT+DMFT spectra by Goremychkin   T coh . DFT+DMFT calculations are reproduced from Goremychkin et al. [5] (online materials), as well as from Sakai [6]. The results of our NCA/AIM "toy model" discussed in Supplementary Note 1 are also shown (Parameter Set 1). Lines drawn in corresponding colors indicate data measured/calculated at similar momentum transfer. Each set of calculations is arbitrarily scaled to the RIXS data, however with the same scale factor for high and low temperature calculations.

Supplementary Note 3: Comparison of RIXS and inelastic neutron scattering
The fact that the RIXS excitations of CePd 3 appear in the crossed polarization channel suggests that the dynamic magnetic susceptibility χ (Q, ω), as measured by INS, may provide a useful comparison. We measured the polycrystalline-averaged dynamical correlation function S(ω) = [1 + n(ω)] 1 π χ (ω) [ n(ω): Bose occupation factor] on at ARCS (ORNL), which allows access to the 50-250 meV range. The spectrum is shown in Supplementary  Figure 3(b) along with data reproduced from Fanelli et al. [3] (measurement at 7 K), Murani et al. [7] (10 K), and Murani et al. [8] (12 K). For reference, the RIXS spectra of Figure 3 of the manuscript are reproduced in Panel (a), and the ranges of these RIXS intensities are also indicated in Panel (b) by gray shaded margins. Where available, the spectra are presented in absolute units.
Remarkably, RIXS and INS are quantitatively consistent up to ∼ 200 meV, i.e., both in the energy position, lineshape, and in the relative intensity between low and high-temperature datasets [3]. On the other hand, RIXS and INS differ markedly at energy transfers above 200 meV. Due to the weakness of dipole transition matrix elements between the J = 5/2 and J = 7/2-like states, excitations across the spin-orbit gap are hardly measurable in INS. For reference, the red markers in Supplementary Figure 3

Overview of the Method
For our computation of Ce M 5 -edge RIXS spectra using the DFT+DMFT scheme, we proceed in several steps [9][10][11][12][13][14][15]. First, a standard DFT+DMFT calculation is performed for the experimental crystal structure of CePd 3 , as follows. The DFT bands within the generalized gradient approximation (GGA) [16] for the exchange-correlation potential are obtained using the WIEN2K package [17], which implements the augmented plane wave and the local-orbital (APW+lo) method. Muffin-tin radii (RMT) of 2.5 r Bohr are used for Ce and Pd atoms. The maximum modulus for the reciprocal vectors K max was chosen such that R MT ×K max = 7.0. The Brillouin zone was sampled in a 20×20×20 k-mesh. The spin-orbit coupling (SOC) is also taken into account in the GGA calculations.
Second, the DFT bands are projected onto a tight-binding model spanning Ce 4f and 5d bands, and Pd 4d, 5s, and 5p bands using the wien2wannier [18] and wannier90 [19] codes. This tight-binding model is augmented by the local electron-electron interaction U within the Ce 4f shell. Following previous DFT+DMFT and spectroscopy studies for Ce compounds [5,20], we employed U = 6 eV for the Hubbard parameter. The bare energy of the Ce 4f states is obtained from the GGA value by subtracting the double-counting correction µ dc , which accounts for the Ce 4f -4f interaction already present in the GGA description. In the absence of a unique definition of µ dc , we treat µ dc as a parameter, adjusted by comparison to experimental valence and inverse photoemission spectra [21,22] (see below / Supplementary Figure 5).
Finally, to compute Ce M 5 -edge RIXS and XAS spectra using the Kramers-Heisenberg formula [23] and Fermi's golden rule, we adopt the configuration-interaction (CI) solver [13,15,24]. This AIM solver uses the hybridization density V 2 (ε) obtained from DMFT. The AIM is augmented by the Ce 3d core states and the 3d-4f interaction present in the intermediate state of RIXS (corresponding to the final state of XAS). The core-hole potential U f c is set to 8.8 eV, and the higher Slater integrals obtained from atomic Hartree-Fock calculation [25] are scaled down to 80 % of their actual values to simulate the effect of intra-atomic configuration interaction from higher basis configurations neglected in the atomic calculation [25][26][27].
As the CI solver works directly in real frequencies, it allows to resolve fine spectral features in the spectra. However, it needs to approximate the continuum hybridization densities V 2 (ε) by a finite set of discrete bath levels. This CI method allows us to include more bath levels than in a standard exact diagonalization algorithm (see recent studies for 3d transition metal oxides [15,24]). Nevertheless, for the case of a Ce 4f impurity with a large number of internal (local) degrees of freedom, the necessarily finite number of bath levels still poses a severe limitation.

Projection of bands onto a local basis
The Wannier functions w(r) representing the Ce 4f states in the tight-binding model are summarized in Supplementary Figure 4. In projecting the Ce 4f bands, we adopt the Γ 6,7,8 basis, i.e. these Wannier functions are constructed directly as spinors (with different up and down components). The spin components of the Ce 4f Wannier functions are displayed in Supplementary Figure 4. The local Hamiltonian at the Ce site can be found in the table shown in this figure, where small off-diagonal matrix elements are allowed between the Γ 7 −Γ 7 and Γ 8 −Γ 8 states.
Since our Ce 4f Wannier functions are an irreducible representation of the cubic symmetry, the off-diagonal hybridization densities V 2 (iω n ) [discussed in Supplementary Note 4.4] are allowed only between the states belonging to the same (two Γ 7 or two Γ 8 ) representations. Given that the two Γ 7 (or Γ 8 ) states are well split by the spin-orbit coupling, the resulting off-diagonal hybridization densities are small [cf. Supplementary Figure 6(c)].
To model the electron-electron interaction in the AIM calculation, these small off-diagonal hybridization densities are neglected, and only the isotropic Hubbard term U is taken into account. Using this computational setup, the CT-QMC simulation can access low temperatures, free from the sign problem.

The double counting parameter µ dc
As stated above, the bare energy of the Ce 4f states is obtained from the GGA value by subtracting the doublecounting correction µ dc , which accounts for the Ce 4f -4f interaction that is already present in the GGA description. In practice, µ dc renormalizes the energy splitting between the Ce 4f states and Pd 4d (and other) bands, which is illustrated in Supplementary Figure 5. As a consequence, the position of the 4f 2 peak (at around 5 eV in the experimental data) is sensitive to µ dc , despite a small change in the Ce 4f occupation (N 4f in the figure). This allows us to fix µ dc ≈ −2.25 eV.
Aside from these valence spectra (Supplementary Figure 5), the obtained Ce 4f self-energies Σ(ε) are also highly sensitive to µ dc , as discussed below in the context of Supplementary Figure 6.
Supplementary Figure 5: The double-counting (µ dc ) dependence of the valence spectra, as calculated by the DFT+DMFT method. (a) At low temperatures (116 K) and (b) at high temperatures (400 K). The experimental photoemission and inverse photoemission spectra are taken from Refs. [21,22].

Self energies Σ(ε)
In DMFT, the electron dynamics on the Ce sites is described by the Anderson impurity model (AIM) [9,10]. The exchange of the electrons between a Ce site and the rest of the crystal (represented by non-interacting auxiliary bath states) is described by the hybridization density V 2 (iω n ) that is determined self-consistently within DMFT. We use a strong-coupling continuous-time quantum Monte Carlo (CT-QMC) impurity solver [28][29][30][31] to compute the self-energies Σ(iω n ) from the AIM.
Aside from the valence spectra (Supplementary Figure 5), the Ce 4f self-energies Σ(ε) also show a strong µ dc dependence, see Supplementary Figure 6. This relates with the physics of a highly asymmetric Anderson impurity model, where the 4f 2 state (upper Hubbard band) lies at around 5 eV, while the 4f 0 state (lower Hubbard band) is close to the chemical potential. In terms of ionic configurations, the f 0 state is rather close to the f 1 ground state, and thus even a small shift of the bare 4f levels (by the double-counting correction µ dc ) yields a large change of E(f 1 ) − E(f 0 ). This also affects the Kondo temperature. Consequently, the thermal evolution of both the low-energy Ce 4f spectra (Supplementary Figure 5) and the self energy (Supplementary Figure 6) depend on µ dc .
Given this µ dc -dependence, we do not implement the charge self-consistency (updating the DFT charge) in our DFT+DMFT calculation. Instead we adjust µ dc as a key parameter to achieve consistency with experimental results. This includes previously reported (direct and inverse) photoemission spectra (Supplementary Figure 5), as well as the spectroscopic results (ARPES, M -edge XAS, RIXS) reported in the manuscript.
After reaching DMFT self-consistency, we analytically continue Σ(iω n ) with the maximum entropy method [32,33] to Σ(ε) in real frequencies ε. Σ(ε) is then used to calculate the one-particle spectral densities A(ε) and hybridization densities V 2 (ε) at real frequencies, as shown in Supplementary Figure 6

Discretization of hybridization densities
As noted above, the AIM solver works directly in real frequencies. This allows to resolve fine spectral features in the spectra, but requires to approximate the continuum hybridization densities V 2 (ε) by discrete levels. In the present study, we modeled V 2 (ε) by 20 bath levels in the discretization scheme [−5 : 1 : −1, −0.08 : 0.02 : 0, 0.25 : 0.25 : 2.5] eV, as illustrated below.
The CI implementation of the present discrete bath gives an accurate 4f occupation N 4f (the deviation from the numerically-exact one estimated by CT-QMC is less than 5%) for the used µ dc value (−2.25 eV) and simultaneously reproduces the energy of the J = 5/2 → 5/2 transitions (∼ 70 meV, see the main text). In testing various discretization schemes of V 2 (ε), we found that only those with an accurate N 4f value also give a good match in the position of the J = 5/2 → 5/2 peak. At high temperatures, low-energy RIXS features are basically atomic excitations involving the J = 7/2 and J = 5/2 manifolds, i.e., the given Ce site and the rest of the crystal are effectively decoupled from each other. Thus the details in the bath discretization do not affect the low-energy RIXS features.
Supplementary Note 5: RIXS incident energy dependence and Pd 4d bands Supplementary Figure 7 shows the variation of RIXS spectra of CePd 3 in the vicinity of the Ce M 5 absorption edge. The measurements were obtained at room temperature, by exciting the sample with x-rays linearly polarized in the scattering plane, and with polarization-resolution of the scattered beam (ππ +πσ polarisation channels). The data allows distinguishing a broad fluorescence-like signal with energy-transfers proportional to the excitation energy (E ∝ ω in ) from the Raman-like (E =const.) interband excitations. The right panel shows a detailed view of low energy transfers. We combine RIXS polarization-analysis and photon-energy (ω in ) dependence to separate the spin-orbit channels of the interband excitations and assign their character. As shown in Supplementary Figure 9(a), at ω in = ω res − 1.5 eV, the incident energy does not suffice to access intermediate states in which Ce f 1 states have been excited to the J = 7/2 state. At ω in = ω res −0.5 eV, as shown in Panel (b), these excitations are strongly enhanced. The polarization analysis of the scattered beam reveals that they appear mostly in the crossed (σπ ) channel. This is further emphasized by subtracting the former from the latter spectra, see Panel (c), middle. The spectral weight attributable to spin-orbit excitations can thus be isolated. In turn, subtracting this signal from the ω res − 0.5 eV data separates excitations within the ground state manifold, see Panel (c), bottom. Overall, both transitions appear with similar spectral weight in the crossed polarization channel. The quasielastic Thompson scattering, as well as a sloping background signal appear in the parallel (σσ ) channel. The M-edge x-ray absorption spectroscopy (XAS) characteristics observed in this study are consistent with earlier fluorescence-yield measurements [34] and resemble those of other strongly valence fluctuating Ce materials [35].

Supplementary
In Supplementary Figure 10, we compare Ce M edge x-ray absorption spectra of CePd 3 obtained at the RIXS instrument [in total electron yield (TEY) mode, as shown in Figure 2(d) of the manuscript] with those obtained in partial-fluorescence-yield (PFY) mode at the MaRES endstation of BL29/BOREAS (ALBA). During the RIXS experiment, TEY-XAS spectra were measured as the drain current from the sample surface (at 22 K). The photon energy range shown here covers both Ce M 5 (3d 5/2 → 4f 7/2 , 884 eV) and M 4 (3d 3/2 → 4f 5/2 , 902 eV) absorption edges, split by the spin-orbit coupling of the 3d 9 core-hole. These TEY-XAS spectra did not vary with the incident angle of the beam, which indicates that the measurement is not strongly affected by surface contamination or reconstruction. The intricate lineshapes reflect the impact of the core-hole potential on the configuration of the largely unoccupied 4f manifold, corresponding to the available intermediate states of the RIXS process. The photon energy hν = 882.2 eV chosen in this study corresponds approximately to the center of the M 5 XAS double-peak in trivalent Ce compounds [34,36,37]. Supplementary Figure 10: Cerium M edge x-ray absorption spectra. Data recorded in total electron yield mode at the RIXS instrument (ID32) is compared with partial fluorescence mode data obtained at BL29/BOREAS (ALBA). The resonance energy at which the RIXS spectra shown in the manuscript were obtained (882.2 eV), is marked by a gray line.